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Theorem ressid 13219
Description: Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Hypothesis
Ref Expression
ressid.1  |-  B  =  ( Base `  W
)
Assertion
Ref Expression
ressid  |-  ( W  e.  X  ->  ( Ws  B )  =  W )

Proof of Theorem ressid
StepHypRef Expression
1 ssid 3210 . 2  |-  B  C_  B
2 ressid.1 . . 3  |-  B  =  ( Base `  W
)
3 fvex 5555 . . 3  |-  ( Base `  W )  e.  _V
42, 3eqeltri 2366 . 2  |-  B  e. 
_V
5 eqid 2296 . . 3  |-  ( Ws  B )  =  ( Ws  B )
65, 2ressid2 13212 . 2  |-  ( ( B  C_  B  /\  W  e.  X  /\  B  e.  _V )  ->  ( Ws  B )  =  W )
71, 4, 6mp3an13 1268 1  |-  ( W  e.  X  ->  ( Ws  B )  =  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165
This theorem is referenced by:  submid  14444  subgid  14639  gaid2  14773  subrgid  15563  rlmsca  15968  rlmsca2  15969  pjff  16628  rlmbn  18794  ishl2  18803  evlrhm  19425  evl1sca  19429  evl1var  19431  pf1ind  19454  dchrptlem2  20520  lnmfg  27283  lmhmfgsplit  27287  pwslnmlem2  27298  dsmmfi  27307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-ress 13171
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